After a quick tour of the tools,

I'LL DO these ... Linear pair is supplementary Distance from angle bisector Construct a square Angle inscribed in a semicircle

YOU DO these ... Triangle Sum is 180 Degrees Pythagorean Theorem Construct a hexagon Construct a dynamic kite in a circle

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Step by step ...

• Illustrating that a linear pair is supplementary:
  
  
  1. In the Display menu, select Preferences; set the Angle unit to Degrees; turn on Autoshow label for points.
  2. Use the segment tool to construct segment AB.
  3. Use the point tool to place a point C on segment AB.
  4. Use the ray tool to construct a ray CD from point C.
  5. Shift-select angle ACD; from the Measure menu select Angle.
  6. Shift-select angle BCD; Measure >> Angle.
  7. Marguee the two angle measurements.
  8. Under the Measure menu select Calculate...
  9. On the calculator, under the Values menu, you will see the angle measures that had been highlighted; select "∠ ACD", then select the "+" button on the calculator, then Values >> ∠ DCB; select OK. The sum will appear.
  10. As you grab-and-drag point D, the individual angle measures change but the sum remains 180 degrees.
  
  
  
• Illustrating that the distance from a point on an angle bisector to either side of the angle is the same.
  
  
  Use the ray tool to construct an ∠ CAB.
  1. Shift-select, in order, points C-A-B (or B-A-C); Construct >> Angle Bisector.
  2. Place point D on the bisector.
  3. Shift-select point D and ray AB; Construct Perpendicular Line.
  4. Create the point of intersection of the perpendicular and the side by either bringing the select arrow close to the intersection and clicking, or by shift-selecting the ray and the perpendicular and using Construct >> Point At Intersection.
  5. Select the perpendicular; Display >> Hide. Note that the point of intersection remains visible on the side of the angle. Construct the segment from this point on the side to D on the angle bisector. Measure the length of the segment.
  6. Repeat this process for the segment joining point D to the other side of the angle. Slide D along the bisector and note what happens.
  7. What type of quadrilateral is AEDF? How could you prove it?
  
  
  
• Constructing a square by using transformations.
  
  
  • Constructing a square using a reflection.
    1. Construct a vertical segment AB.
    2. Construct a circle with center A and radius AB.
    3. Construct the perpendicular to segment AB passing through point B.
    4. Construct C as one of the points of intersection of the perpendicular and the circle. Hide the perpendicular and the circle. Use the segment tool to construct triangle ABC.
    5. Select segment CA; Transform >> Mark Mirror (you'll see the select sqaure flas as bullseyes).
    6. Use Edit >> Select All, or drag a marquee to select all segments and points; Transform >> Reflect to create the square.
    7. Measure the lengths of segments AB and AC. Calculate the ratio of AC/AB. What is this value?
    
    
    
  • Constructing a square using rotations.
    1. Construct a vertical segment AB.
    2. Double-click on point A so that it flashes as a bullseye. This shows it was marked as the center of a rotation. You can also select it and use Transform >> Mark Center.
    3. Select all. Transform >> Rotate. A dialog box comes up that allows you to specify the angle of rotation. Use 90 °.
    4. Construct segment BB'. Select this segment and Construct >> Point at Midpoint.
    5. Mark the midpoint as the center. Select all, Tranform >> Rotate by 180 °.
  
  
  
• Showing that any angle inscibed in a semicircle is a right angle.
  
  
  1. Let Circle (A, AB) specify the circle with center point A and radius AB. Construct circle (A, AB).
  2. Construct ray BA. Identify the point of intersection of the ray and the circle. Hide the ray, and construct the segment. This will create the diameter.
  3. Construct a random point on the circle. Connect that point to the endpoints of the diameter.
  4. Measure the angle whose vertex is the random point on the circle.
  5. Just for fun, shift-select the point and the circle; Display >> Animate slowly.